DATA SCIENCE SESSIONS VOL. 3

A Foundational Python Data Science Course

Session 15: Regularization in MLR. The Maximum Likelihood Estimation (MLE).

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Feedback should be send to goran.milovanovic@datakolektiv.com.

These notebooks accompany the DATA SCIENCE SESSIONS VOL. 3 :: A Foundational Python Data Science Course.

Lecturers

Goran S. Milovanović, PhD, DataKolektiv, Chief Scientist & Owner

Aleksandar Cvetković, PhD, DataKolektiv, Consultant

Ilija Lazarević, MA, DataKolektiv, Consultant

1. Regularization in Multiple Linear Regression

1.1 Lasso (L1), Ridge (L2), and Elastic Net regression

Regularization is used for several purposes:

Consider the following:

$$Loss = \sum_{i=1}^{n}{\big(y_i-\hat{y_i}\big)^2}$$

It is just the ordinary sum of squares, or the loss (or cost) function of the Linear Regression Model. The parameters $\beta$ of the Linear Model are estimate by minimizing the above quantity - the model's cost function. The expression above is just the ordinary $SSE$ function as we have encountered it before in our sessions on Linear Regression.

Now, consider the following formulation of the cost function that includes the penalty term:

$$Loss_{L2} = \sum_{i=1}^{n}{\big(y_i-\hat{y_i}\big)^2}+\lambda\sum_{j=1}^{p}\beta_j^2$$

$\lambda\sum_{j=1}^{p}\beta_j^2$ is the penalty term: it increases the value of the cost function by a factor determined by the sum of the squared model coefficients. In the expression above $p$ stands for the number of model parameters (coefficients, in the case of Linear Regression) and $\beta$ are the coefficients themselves. The above expression represents the $L2$ regularization for the Linear Model, also known as Ridge Regression. If $\lambda$ is zero we get back to the Ordinary Least Squares model that we have already learned about. If $\lambda$ is too large, it will add too much weight to the cost function which will lead to underfitting.

The underlying idea is to penalize large coefficients: it regularizes the coefficients so that if they take large values the cost function is penalized. In effect, the Ridge regression shrinks the coefficients so to reduce the model complexity and reduces the effects multicollinearity.

Consider now the following modification of the cost function:

$$Loss_{L1} = \sum_{i=1}^{n}{\big(y_i-\hat{y_i}\big)^2}+\lambda\sum_{j=1}^{p}|\beta_j|$$

The above cost function is based on a weighted sum of the absolute values of the model coefficients. This approach is known as $L1$ or Lasso Regression.

The most important difference between Ridge and Lasso regularization is that in Lasso some coefficients can be shrinked to zero and get completely eliminated from the model. Thus, Lasso regularization does not only prevent overfitting but also performs feature selection - a very handy thing when considering large linear models.

Finally, consider the following formulation of a loss (or cost) function:

$$Loss_{ElasticNet} = \frac{\sum_{i=1}^{n}(y_i-\hat{y_i})^2}{2n}+\lambda(\frac{1-\alpha}{2}\sum_{j=1}^{m}\hat{\beta_j}^2+\alpha\sum_{j=1}^{m}|\hat{\beta_j}|)$$

In the expression above - the Elastic Net regression - $\alpha$ is the mixing parameter between Lasso ($\alpha=1$) and Ridge ($\alpha=0$) approach. While the Ridge regression shrinks the coefficients of correlated predictors towards each other, the Lasso regression tends to pick one of them and decrease the others. The Elastic Net penalty is a mixture of the two approaches controlled by $\alpha$.

1.2 Inspect Multicolinearity in one particular Linear Regression problem

Target: predict price from the numerical predictors

1.3 Regularization of the model coefficients: Ridge (L2) Regularization

Now we are going to use scikit-learn to perform Regularized Multiple Linear Regression; we'll be using the Ridge Regression.

As we have see, the Ridge Regression model is obtained by minimizing the function

$$SSE + \alpha L_2^2,$$

where $SSE$ is the Sum Squarred Error of the 'ordinary' Multiple Linear Regression

$$\hat{y} = \beta_1x_1 + \beta_2x_2 + \cdots + \beta_kx_k + \beta_0,$$

and $L_{2}^2$ is the squared $L_{2}-$ norm of Multiple Linear Regression model parameters $\beta_1,\beta_2,\ldots,\beta_k$:

$$L_2^2 = \beta_1^2 + \beta_2^2 + \cdots + \beta_k^2,$$

and hence the name L2 Regularization. $\alpha$ is called penalization (hyper)parameter, and it's used to control the magnitude of models parameters; for $\alpha = 0$ we recover ordinary MLR.

We see that the best model is obtained with small penalization hyperparameter, i.e. with large model parameter (coefficient) values.

1.4 LASSO (L1) Regularization

LASSO is another way to regularize MLR model. It's obtained by minimizing the function

$$SSE + \alpha L_1,$$

where $SSE$ is Sum Squared Error of the ordinary Multiple Linear Regression and $L_1$ is $L_1-$ norm of its coeffitents given by

$$L_1 = |\beta_1| + |\beta_2| + \cdots + |\beta_k|,$$

and hence the name L1 Regularization. $\alpha$ is, of course, penalization hyperparameter.

We see that the best model is obtained with small penalization hyperparameter, i.e. with big model parameters values.

2. Maximum Likelihood Estimator (MLE)

Tossing a coin

Imagine I have a fair coin: the one that lands Heads and Tails with equal probabilities P(H) = P(T) = 1/2.

Uhm, but I have already assumed that we will be tossing a fair coin? Four heads, six tails, how come? Is this really a fair coin? The answer is: it still might be. Let's see.

Ok, now let' see what happens with 10,000 statistical experiments with a fair coin:

What about 100,000 experiments?

Theoretical probability: when we know that some stochastic system produces an outcome with some particular probability.

Experimental probability: when we estimate that some stochastic system produces an outcome with some particular probability from observed data.

Reverse engineering the coin

In all my previous statistical experiments, I have assumed that the coin is fair, e.g. P(H) = P(T) = .5, look:

tosses = np.random.choice(outcomes, 100, replace=True, p=[.5, .5])

I have used numerical simulations of coin tosses to demonstrate that with increasing number of experiments I get to estimate the known probabilities of P(H) and P(T) better and better.

However, we typically do not know the theoretical probability of a random event... Example: the CTR of some social media post.

CTR (Click-Through Rate) is like a coin toss: a post is clicked or not clicked upon each served impression. We can thus threat the CTR like a coin toss: P(H) = P(Clicked) is "post clicked", P(T) = P(Not clicked) is "not clicked". Then we can say that P(Clicked) is some latent, essential characteristic of post.

Imagine we have a set of observations for a particular social media post: 1,1,1,0,0,0,1,0,1,0,1,0,0,0,1,0.., where 1 = Clicked and 0 = Not clicked. How do we estimate the CTR of the post from these data?

Easy, right: sum up the vector of observations and divide by its length. But wait. There's more to it.

I have just invented some social media post with a CTR of .3, and estimated its experimental probability to be .3035; that is the best estimate that I currently have. What if I want to test a set of hypotheses about the CTR of this post? What, for example, if I am intersted to learn about the probability of this post having a CTR of .25 instead? Let's use a simpler, smaller set of observations to begin with.

Imagine we toss a fair coin with P(H)=.5 twice and observe two Heads. The probability of observing two heads with P(H)=.5 is:

.5 * .5 = .25

But what if the coin was not fair, and instead had a probability of P(H)=.3? Then:

.3 * .3 = 0.09

And what if the coin had P(H) = .7?

.7 * .7 = 0.49

With P(H) = .9 we get:

.9 * .9 = 0.81

and it seems intuitive and logical: if we have observed a sample - however small it was - of all heads (and are assuming that we have observed two Heads from two tosses here), it makes sense that we are dealing with a coin that almost always lands Heads.

Let's express this intuition visually. First, we need a function to compute the probability that a coin with a certain P(H) produced the data.

Nice. Now:

Let's ask explicitly:

And what if P(H) was .85?

Pick any arbitrary observation

Good. What have we learned?

To estimate the parameter of a simple statistical model from data - a coin, in our case, a simple stochastic system with two outcome states only - we need to find the maximum of its likelihood function given the observed data.

For our example with the HH minimal observations - tossing a coin two times and observing two Heads in a row - the likelihood function is:

Of course: however small the sample size, the sample represents just everything that we know about a coin; if we observe all heads, like in: HH, then the most probable coin that could have produced the result is definitely the one that yields Heads all the time!

2.1 MLE in Linear Regression

Further Reading

DataKolektiv, 2022/23.

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